Resumen de The cutoff profile for the simple exclusion process on the circle

Hubert Lacoin

• In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let PtPt denote the semi-group associated the exclusion on the circle with 2N2N sites and NN particles. For any initial condition χχ, and for any t≥4N29π2logNt≥4N29π2log⁡N, we show that the probability density Pt(χ,⋅)Pt(χ,⋅) is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As 4N29π2logN4N29π2log⁡N is smaller than the mixing time which is N22π2logNN22π2log⁡N, this allows to give a sharp description of the cutoff profile: if dN(t)dN(t) denote the total-variation distance starting from the worse initial condition we have limN→∞dN(N22π2logN+N2π2s)=erf(2–√πe−s), limN→∞dN(N22π2log⁡N+N2π2s)=erf⁡(2πe−s), where erferf is the Gauss error function.